Question: A circle has a circumference of $12\pi$. It has an arc of length $\dfrac{32}{5}\pi$. What is the central angle of the arc, in degrees? ${12\pi}$ ${192^\circ}$ $\color{#DF0030}{\dfrac{32}{5}\pi}$
Answer: The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{s}{c}$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{32}{5}\pi \div 12\pi$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{8}{15}$ $\theta = \dfrac{8}{15} \times 360 ^ \circ$ $\theta = 192^\circ$